ALMA Memo 395
Alignment Tolerances for ALMA Optics
B.Lazareff
IRAM
25-Oct-2001
Abstract
A tolerance analysis of the alignment of the ALMA FE optics is presented.
The following performance criteria are taken into account: a) loss of onaxis efficiency from aperture plane misalignment; b) focal plane coalignment of the two beams of each band; c) aberrations when off-axis
mirrors operate between wavefronts not centered on the foci of the
ellipsoid; d) non-uniform phase across the primary beam resulting from
aperture plane misalignment.
It is found, based on reasonable machining tolerances, and existing FEM
simulations of pressure bending, that the adopted goals can be met, the
driving goals being on-axis efficiency (a) and primary beam phase (d).
The present study, especially in view of the fact that detailed designs of
cartridge internal optics do not exist at the time of writing, can only be
considered as a guideline. The author cannot take responsibility for the
fact that a particular cartridge will or will not meet alignment specs, based
on, e.g., tolerances written on fabrication blueprints that do not yet exist.
Some assumptions had to be made concerning accuracies used as input in
the calculation, that need to be confirmed by the groups respectively in
charge.
1.
Introduction
Through the efforts of several ALMA teams, an optical configuration has been designed with the goal
of optimizing its efficiency, taking into account requirements or constraints from other areas, notably
cryogenics. The geometrical optical design is about to be finalized. Due to small fabrication errors and
environmental factors (gravity, thermal, etc), the front-end as-built will inevitably deviate from the
ideal design. It is therefore necessary, before the detailed design is undertaken, to quantify the impact
of deviations between the actual and designed optics against the system performance specifications.
Then, a tolerance budget can be divided up between subsystems and components, so that fabrication
techniques and control procedures can be designed to meet the system specifications.
For the purpose of the present analysis, the reference optical design is the one described in ALMA
Memo 362 (Lamb et al) [1]. Fruitful discussions with P. Napier, J. Lamb, R. Lucas, and S. Guilloteau
are acknowledged.
2.
System specs and tolerances: outline
2.1. Specifications
The present study is driven by two specifications, neither of which is explicitly part of the project
specifications:
1 / 14

Minimal loss of optical efficiency of the actual front-end, compared with the design. A nominal
criterion of 1% efficiency loss is used.

Deviations from flat phase in primary beam consistent with the requirement for amplitude errors:
pointing accuracy 6% of HPBW, as specified in Table 2.1, Chapter 2 of the ALMA Project
Book [2], and resulting in an antenna specification of 0.6" pointing error. Note that the pointing
accuracy refers to random errors, while the primary beam phase pattern, in principle, is stable and
can be calibrated.
These will be discussed in more detail further down.
2.2. Impact of misalignments
Mechanical misalignments cause the parameters of the beam illuminating1 the secondary to deviate
from their nominal values. Such deviations will be reckoned in the telescope focal plane and can be
classified as:
1) Displacement
a) Along telescope axis
b) Lateral shift in focal plane (= tilt in aperture plane, = pointing offset on the sky)
c) Tilt in focal plane (= lateral shift in aperture plane, =loss of aperture efficiency)
d) Rotation about telescope axis
2) Distortion (coupling to higher order modes, if the launched beam is fundamental gaussian) within
the front-end optics.
Effect 1a is not considered here, because even in the nominal design, the various bands are not
constrained to have a common focus, as long as the amount of spherical aberration associated with
refocusing is tolerable (see Memo 362). Effect 1b is considered only insofar as concerns the coalignment of the two orthogonally polarized beams of one band; the common pointing offset needs to
be calibrated for each band anyway. Effect 1c is the dominant one in the present study. Effect 1d is
weighed by the sine of the small angle between each beam and the telescope axis, and can be ignored
if a reasonable level of general tolerance is maintained. Effect 2 will be addressed, but it will be found
that it is not driving the tolerances.
2.3. Interplay of misalignments and specifications
This can be summarized graphically:
Co-Align.
orth. pols.
Beam
Displacement
Calibration
Errors
Beam
Distortion
Efficiency
Loss
Misalignment
Figure 1: Flow diagram from misalignements to optical properties
3.
Efficiency loss from beam deviation
Based on a gaussian illumination with a 12dB edge taper, one finds that a 1% drop of on-axis
efficiency is reached when the illumination pattern is offset from the center of the secondary by
1
As is common practice, I regard the optics and telescope working as a transmitter.
2 / 14
0.088  rs . This corresponds to an angular offset of 5.5mrad from the nominal launch angle in the
focal plane. I rounded this up to 6mrad (1.2% efficiency loss). The actual edge taper found from the
electromagnetic analysis of Tham and Withington [3] (P.O. analysis available for bands 3–6, and 9 at
the time of writing) lies between 10.5 and 11.9dB; it was felt that the discrepancy did not warrant a reevaluation of the tolerance criterion.
I use the ABCD equations of geometrical optics to propagate the perturbation of the chief ray; one can
show that the same equations apply to the main axis of a gaussian beam. It is assumed that the
unperturbed optical path is contained in a plane, and only perturbations of the chief ray within that
plane are computed, which simplifies the work while still providing a first approach at an estimate of
tolerances.
A summary of the method is given in Appendix A.
3.1. Results: Sensitivity to individual misalignments
In a first step, the sensitivity of the position to individual misalignments is computed. Both linear and
angular misalignments are considered, in the plane of the folded optical path (or in an arbitrary plane
for bands 1 & 2). Also two types of misalignments are considered:

Independent displacement of an optical element;

A "break", i.e., all elements from the feed up to some element in the optical train are perturbed, the
other elements remaining unperturbed. This covers, as a particular case, the global misalignment
of a cartridge (bands 5-10).
Although both the lateral and angular displacements of the beam at the focal plane are computed, the
former is so small that it is negligible as concerns the aperture plane alignment, and only the angular
displacement is considered in this part of the analysis. The results are summarized in graphical form in
figures 2 and 3.
Lin. Tol. / Ap. Illum.
1.2
BkFeed
Tol (mm)
1
Mirr1
Bk1
0.8
Mirr2
Grid
0.6
0.4
0.2
0
1
2
3
4
5
Band
6
7
8
9
10
Figure 2: Tolerances for linear displacements of optical elements that
would alone cause a 6mrad misalignment of the beam. Legend: BkFeed:
"break" after feedhorn, synonymous with feedhorn displacement; Mirr1:
displacement of mirror/lens #1; Bk1: "break" after lens/mirror #1 (see
Figure A-1 in Appendix 1 for definition of a break); Mirr2: displacement of
mirror 2; Grid: displacement of grid (if applicable).
3 / 14
Ang. tol. / Ap. illum.
Tol (mrad)
10
9
8
Mirr1
Bk1
Mirr2
7
6
5
4
3
Bk2
Grid
2
1
0
1
2
3
4
5
6
7
8
9
10
Band
Figure 3: Tolerances for angular displacements of optical elements that
would alone cause a 6mrad misalignment of the beam
3.2. Results: Global tolerance budget.
To complement the analysis presented above, I now perform a global, forward, tolerance evaluation.
Reasonable (?) guesstimates are made for the accuracy with which various elements can be positioned
with respect to each other, and estimates are derived for the angular deviation of the beam "emitted"
by the receiver towards the secondary. In addition to the misalignments already considered, I also take
into account the following:

The cold optics elements are assumed to be mounted on some "sub-frame", which itself has a
positioning accuracy w/r to the cartridge's 4K plate;

The displacement of the 4K plate w/r to the vacuum vessel, including the pressure, gravity, and
thermal deformation, and the machining accuracy;

The misalignment, due to machining accuracy, of the back plate of the vacuum vessel (to which
the cartridge is referenced), w/r to the front side interface of the vacuum vessel to the telescope;

The misalignment between the reference frame defined by the telescope interface to the vacuum
vessel, on one hand, and the line of sight to the center of the subreflector, on the other hand.
The displacement of the 4K plate w/r to the vacuum vessel due to pressure deformation of the latter is
estimated from a map of deflections of the back plate, kindly provided by M.Harman (Ansys FEM
calculation performed Apr 5, 2001).

Bands 1 and 2: differential sag of the back plate: z = 0.046 mm; cartridge diameter D = 170 mm;
angular deflection:  = 0.27mrd; resulting lateral motion at distance H = 600 mm from base
x = 0.16 mm.

Bands 3 and 4: differential sag of the back plate: z = 0.080 mm; cartridge diameter D = 140 mm;
angular deflection:  = 0.57mrd; resulting lateral motion at distance H = 600 mm from base
x = 0.34 mm.

For the other bands, a value of  = 0.5mrd is used (not critical), while the linear offset is not
significant because it affects the FE optics globally and amounts to a pointing offset.
The gravity deformations of the current baseline cartridge (tubular GFRP support structure, no holes)
are negligible: 2.5m linear, 8rad.
The machining accuracy in positioning the 4K plate w/r to the vacuum vessel is guesstimated at
0.05mm, 0.2mrad.
4 / 14
The assumed linear and angular tolerances are multiplied by the appropriate elements of the P matrix
(see above and Appendix A), and two values of the global misalignment are computed:

Worst case: sum of absolute values;

RSS: square root of sum-of-squares.
The geometry of the tolerancing is still only 2-D. A pessimistic estimate might be obtained by
multiplying the global misalignment by
2 . The results are listed in Table 1.
Table 1. Estimates of the angular misalignment of the beam coupled by the
FE optics. Both worst-case and RSS estimates are given for each band.
Assume some standard values for tolerances of:
Value
Unit
Var.
Position of cartridge elements within cartridge optics
Linear
0,02 mm
CEL
Angular
1,0E-04 rd
CEA
Global position of cartridge optics w/r to 4K plate
Linear
0,02 mm
COL
Angular
1,0E-04 rd
COA
Position of cartridge 4K plate w/r to vacuum vessel
Pressure
Machining
Bnd 1-2 Linear
0,16 mm
0,05 mm
Angular
2,7E-04 rd
2,0E-04 rd
Bnd 3-4 Linear
0,34 mm
0,05 mm
Angular
5,7E-04 rd
2,0E-04 rd
Other
Linear
0,25 mm
0,05 mm
Angular
5,0E-04 rd
2,0E-04 rd
Positition of vacuum vessel w/r to receiver flange
Angular
5,0E-04 rd
VFA
Position of receiver flange w/r to true telescope axis
Angular
5,0E-04
FTA
Total
0,21 mm
4,7E-04 rd
0,39 mm
7,7E-04 rd
0,3 mm
7,0E-04 rd
CVL1
CVA1
CVL3
CVA3
CVL
CVA
Note: for bands 1-4, the global position error of the cartridge optics w/r to the vacuum vessel
introduces an internal break in the optical train. For bands 5-10, it introduces a global
misalignment of the receiver beam. Therefore, the tolerance equations are different for
these two groups of bands.
For each band, the coefficients (partial derivatives) of beam exit angle versus misalignment
are listed; then the individual contributions; finally the estimates "worst-case" and "rss".
Note: A final factor of SQRT(2) has not been applied.
Band 1
Lin.
Ang
Coeff
From
Contrib
rd/mm
Coeff
From
Contrib
rd/rd
WorstCase
RSS
rd
Horn
Horn
Horn
Global
5,3E-03 5,3E-03 5,3E-03
CEL
COL
CVL1
1,1E-04 1,1E-04 1,1E-03
rd
0
Global
1
1
VFA
FTA
5,0E-04 5,0E-04
2,3E-03rd
1,3E-03rd
5 / 14
Not Used
Band 2
Lin.
Ang
Coeff
From
Contrib
rd/mm
Coeff
From
Contrib
rd/rd
WorstCase
RSS
Band 3
Lin.
Ang
Band 5
Lin.
Ang
Ang
Global
1
1
VFA
FTA
5,0E-04 5,0E-04
3,5E-03rd
2,2E-03rd
rd/mm
Coeff
From
Contrib
rd/rd
rd
rd
Horn
Mirr1
Mirr2
Global
6,7E-03 6,7E-03 6,7E-03 6,7E-03 0,0E+00
CEL
COL
CVL3
CEL
CEL
1,3E-04 1,3E-04 2,6E-03 1,3E-04 0,0E+00
Global
0,024
0,024
0,024
2
2
1
1
CEA
COA
CVA3
CEA
CEA
VFA
FTA
2,4E-06 2,4E-06 1,8E-05 2,0E-04 2,0E-04 5,0E-04 5,0E-04
4,4E-03rd
2,7E-03rd
Essentially identical to band 3
Coeff
From
Contrib
rd/mm
Coeff
From
Contrib
rd/rd
WorstCase
RSS
Band 6
Lin.
0
rd
Coeff
From
Contrib
WorstCase
RSS
Band 4
rd
Horn
Horn
Horn
Global
1,1E-02 1,0E-02 1,0E-02
CEL
COL
CVL1
2,2E-04 2,0E-04 2,1E-03
rd
rd
Global
Global
Global
0,0088
2,16
2
1
1
1
1
CEA
CEA
CEA
COA
CVA
VFA
FTA
8,8E-07 2,2E-04 2,0E-04 1,0E-04 7,0E-04 5,0E-04 5,0E-04
3,5E-03rd
1,3E-03rd
Coeff
From
Contrib
rd/mm
Coeff
From
Contrib
rd/rd
WorstCase
RSS
Horn
Mirr1
Mirr2
Global
0,018
0,031
0,015
CEL
CEL
CEL
3,6E-04 6,2E-04 3,0E-04
rd
rd
Horn
Mirr1
Mirr2
Global
0,023
0,038
0,017
CEL
CEL
CEL
4,6E-04 7,6E-04 3,4E-04
Global
Global
Global
0,01
2,79
2
1
1
1
1
CEA
CEA
CEA
COA
CVA
VFA
FTA
1,0E-06 2,8E-04 2,0E-04 1,0E-04 7,0E-04 5,0E-04 5,0E-04
3,8E-03rd
1,4E-03rd
6 / 14
Band 7
Lin.
Ang
Coeff
From
Contrib
rd/mm
Coeff
From
Contrib
rd/rd
WorstCase
RSS
Band 8
Lin.
Ang
Ang
Coeff
From
Contrib
rd/rd
rd
rd
rd/mm
Coeff
From
Contrib
rd/rd
Coeff
From
Contrib
WorstCase
RSS
Global
Global
0,006
2,07
0,976
2
1
1
1
1
CEA
CEA
CEA
CEA
COA
CVA
VFA
FTA
6,0E-07 2,1E-04 9,8E-05 2,0E-04 1,0E-04 7,0E-04 5,0E-04 5,0E-04
Horn
Grid
Mirr1
Mirr2
Global
0,019
0,019
0,037
0,021
CEL
CEL
CEL
CEL
3,8E-04 3,8E-04 7,4E-04 4,2E-04
Global
Global
Global
0,008
0,66
3,36
2
1
1
1
1
CEA
CEA
CEA
CEA
COA
CVA
VFA
FTA
8,0E-07 6,6E-05 3,4E-04 2,0E-04 1,0E-04 7,0E-04 5,0E-04 5,0E-04
4,3E-03rd
1,5E-03rd
Coeff
From
Contrib
Band 10
Lin.
Coeff
From
Contrib
Global
4,3E-03rd
1,5E-03rd
rd/mm
WorstCase
RSS
Ang
rd
Coeff
From
Contrib
WorstCase
RSS
Band 9
Lin.
rd
Horn
Mirr1
Grid
Mirr2
Global
0,027
0,037
0,024
0,013
CEL
CEL
CEL
CEL
5,4E-04 7,4E-04 4,8E-04 2,6E-04
rd
rd
Horn
Grid
Mirr1
Mirr2
Global
0,039
0,039
0,064
0,039
CEL
CEL
CEL
CEL
7,8E-04 7,8E-04 1,3E-03 7,8E-04
Global
Global
Global
0,004
1,63
4,43
2
1
1
1
1
CEA
CEA
CEA
CEA
COA
CVA
VFA
FTA
4,0E-07 1,6E-04 4,4E-04 2,0E-04 1,0E-04 7,0E-04 5,0E-04 5,0E-04
6,2E-03rd
2,2E-03rd
Horn
rd/mm
rd
rd/rd
rd
Grid
Mirr1
Mirr2
Global
0,043
0,026
CEL
CEL
CEL
CEL
4,0E-04 4,0E-04 8,6E-04 5,2E-04
0,02
Global
Global
Global
0,02
0,007
0,67
3,3
2
1
1
1
1
CEA
CEA
CEA
CEA
COA
CVA
VFA
FTA
7,0E-07 6,7E-05 3,3E-04 2,0E-04 1,0E-04 7,0E-04 5,0E-04 5,0E-04
4,6E-03rd
1,6E-03rd
Evaluation of results.
For bands 1–4, the most important single cause of misalignment is between the cartridge and the
vacuum vessel, which introduces an internal break in the optical train. For higher frequency bands, and
with the assumed values, there are comparable contributions from internal cartridge alignment and
cartridge/dewar/telescope alignment. Band 9 has a higher estimate of misalignment because of the
relatively fast optics that it uses.
The model used for internal cartridge optics tolerancing may be overly simplistic, in that it assumes
that there is just one intermediate support part between any optical element and the 4K plate. A more
7 / 14
complex construction might still be consistent with the above error estimate if the attachment
points/planes of individual elements would be machined on a permanently assembled support
structure, thereby eliminating individual tolerances in the mechanical chain. The pressure deformation
of the vacuum vessel might be compensated in the construction, either a priori from FEA modeling, or
after evaluation of a prototype; but this is not essential to meet the goals, as will be seen below.
All individual tolerances used as input in the calculation need to be confirmed by the respective groups
in charge.
Worst-case or RSS? If the analysis would apply to a single instrument, the worst-case analysis would
maybe be more appropriate. However, the telescopes will operate most of the time in an array, and the
figure of merit is the array efficiency. Since the efficiency of an individual antenna degrades
quadratically with the aperture plane offset, the effective apertures for two antennas and the
corresponding baseline can be written respectively as:
2

 i  

Ai  A0 1  k  
i  1,2

0  



 1  2  2  
2 
A12  A1 A2  A0 1  k  1
 2  2  
0



where  0 is the misalignment for which the reference degradation k is reached. The expectation value
of the baseline effective aperture is:
A12


 A0 1  k  RSS

 0




2



where I have made the implicit assumption that the individual positioning tolerances are in fact rms
values, which is pessimistic if they are actually maximum values (as they should be for mechanical
tolerances).
This said, if we examine the RSS misalignment values for each of the ten bands, and multiply them by
2 to account for the two possible misalignment directions, we can see that they are safely below the
critical value 0  5.5 mrad .
4.
Co-alignment of orthogonal polarizations in focal plane
In this part, using the same estimates for linear and angular positioning errors of individual elements,
and again using the matrices P (see Appendix A) I estimate the misalignment in the focal plane of the
two beams of one channel. This concerns only bands 7 and above, since bands 1-6 are planned to use
OMT's, that in principle guarantee the co-alignment of the two orthogonal polarizations. Remember
that in this work, I consider only the propagation of the fundamental Gaussian mode; physical optics
calculations by Tham and Withington do show small offsets between orthogonally polarized beams
propagating through off-axis mirrors. The results are listed in Table 2.
8 / 14
Table 2. Estimates of the lateral misalignment of the two orthogonally
polarized beams, for each band, expressed in the focal plane, worst-case
and RSS, in mm and as a fraction of the FWHM at mid-band.
Band 7
Lin
Horn
Coeff
mm/mm
From
CEL
Contrib mm
Ang
Coeff
Mirr1
0.91
CEL
0.076
54
218
24
CEA
CEA
0.101mm
Band 8
Lin
mm/mm
From
Coeff
Grid
1.49
CEL
Contrib mm
Ang
0.0298
0.03
53
159
CEA
CEA
Contrib mm
0.0053
WorstCase
0.081mm
RSS
0.045mm
Band 9
Lin
mm/mm
From
Coeff
Grid
2
CEL
Contrib mm
Ang
0.0159
0.015beam
0.008beam
Horn
Coeff
1.5
CEL
mm/rd
From
2
CEL
0.04
mm/rd
0.04
25
From
CEA
Contrib mm
CEA
0.0218 0.0024
0.025beam
0.014beam
Horn
Coeff
CEL
0.06
Contrib mm
0.0054
WorstCase
0.184mm
RSS
3.8
0.0182
mm/rd
From
Grid
3
134
CEA
0.0025
0.0134
WorstCase
0.096mm
0.026beam
RSS
0.058mm
0.016beam
Band 10
Lin
Horn
Coeff
mm/mm
From
CEL
Contrib mm
Ang
Coeff
From
Grid
1.8
CEL
0.036
mm/rd
0.036
48
CEA
Contrib mm
0.0048
WorstCase
0.093mm
RSS
1.8
0.054mm
159
CEA
0.0159
0.034beam
0.019beam
The co-alignment of the two beams of one band remains — under the adopted hypotheses — always
better than 3.5% of the FWHM.
5.
Aberrations introduced by misalignments
With nominal alignment, the off-axis elliptical mirrors couple (almost) exactly the phase of the
incident and reflected fundamental Gaussian beams; losses (i.e., coupling to higher order modes) come
from the mismatch of the amplitude (and polarization) patterns. Under nominal conditions, the centers
9 / 14
1
1
0.05
0
0.05
0
0.05
0.5
0.5
0
0.1
0.05
0
0.05
0
0
0
0
0
0.05
0
0.5
0.05
0.1
0
0
0.05
0
0.05
0.5
0.1
0.05
0.05 0
0.05
0.05
0.15
0.1 0.05 0
0.05 0.1
0.15
1
1
1
0.5
0
0.5
1
1
Delta
Delta
mm
mm
0.5
0
0.5
1
Figure 4. Examples of phase error patterns for misalignments of the
incident wavefront center of curvature, resp. in the plane of reflection,
and in the perpendicular direction. The contours are labeled with the
path error (mm); the mirror coordinates are arbitrary.
of curvature of the incident and reflected wavefronts coincide with the foci of the ellipsoid. This is
equivalent to the condition of equal optical path for all contributing rays.
In a misaligned system, that condition is not met. For a given position of the incident wavefront's
center of curvature, a reflected center of curvature can be defined such that the optical path is constant
(versus the reflection point on the mirror) up to tilt and focus terms; it is simply the paraxial
geometrical optics image. Higher order residuals induce a loss of amplitude (and power) for the
propagation of the fundamental Gaussian mode.
For a given lateral misalignment of the incident center of curvature with respect to the first focus of
the ellipsoid, one can compute the phase error across the surface of the mirror (dominated by
astigmatism).
Two approaches can be taken to estimate the amplitude loss of the reflected beam; they are briefly
described in Appendix B. Here, I adopted the most severe one, based on the amplitude-weighted rms
phase error.
The results are given in Table 3, giving for each band, and for each mirror, the degree of lateral
misalignment leading to a power loss of 1% in the fundamental mode.
Table 3. Amount of lateral misalignment inducing a power loss of 1% in the
fundamental mode, through phase aberrations.
Band
4
6
7
8
9
10
Allowed misalignment (mm)
Mirror 1
Mirror 2
13
N/A
2.4
4
1.9
5.7
2.6
1.7
0.65
0.54
1.3
0.67
These values are much larger than either achievable tolerances or critical individual tolerances derived
in section 3.1 and displayed in figures 2 and 3.
6.
Phase across the primary beam
The discussion in this section stems from a concern expressed by P.Napier. Since then, it seems that
the issue might be resolved by suitable calibration and data processing [S.Guilloteau, private
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communication]; however, that is not officially established, therefore I felt it would be appropriate to
give a presentation of the argument.
An elementary interferometric observation provides an estimate (at one point in the UV plane) of the
Fourier transform of the product of the sky brightness and the voltage patterns of the two antennas of
one baseline. In most, if not all, of radio interferometric work so far, after the raw sky brightness is
reconstructed from the UV data, it is corrected for the power pattern, often using a Gaussian ansatz for
a "typical" power pattern of an antenna of the array. The phase across the primary beam is assumed to
be constant. The latter is true provided the (complex) aperture illumination is azimuthally symmetric.
Conversely, an aperture plane misalignment will result in a non-uniform phase across the primary
beam, leading to imaging errors.
I have performed a numerical simulation of the voltage pattern for a Gaussian illumunation, having a
12dB taper, and offset by 10% of the aperture radius. The phase of the amplitude pattern is close to
0.1rd at the half-power points. If we denote by a the amplitude pattern, normalized to 1 at boresight,
and a0 the same with flat phase, the error in apparent amplitude a  a 0 , is maximum close to the halfpower points ( a 0  2 1 2 ), where a  a0  0.07 . Since the effect is clearly linear versus the aperture
plane offsets, these numerical results can be generalized to :
3dB  x s / rs
a  a0 3dB  a0,3dB  3dB
where x s / rs is the normalized aperture plane misalignment.
Following a remark by R.Lucas, the errors in the imaginary part of the apparent amplitude — resulting
from aperture plane misalignment — can be compared with the errors in the real part — resulting from
focal plane, or pointing errors. Both should play the same role in imaging, in particular mosaicing.
The technical requirement for pointing, flowing from the scientific requirement for high fidelity
imaging, is that the pointing accuracy should be 6%HPBW (Project Book, Chap 2, Table 2.1). The
resulting antenna performance requirement is 0.6" pointing accuracy (Chap 2, Table 4.1). For a
pointing error that is a fraction f of the HPBW, the error in the apparent amplitude a of a point source
observed at the half-power point is:
a  2 Ln(2) f  0.98 f  f
Based on the scientific requirement, f = 0.06, this leads to a  0.06 , consistent with the magnitude in
the imaginary part of a resulting from the spec x s / rs  0.088 derived in section 3 of the present
report, from the consideration of minimal loss of on-axis efficiency.
However, the actual antenna built has an absolute pointing accuracy, independent of wavelength,
which is a smaller fraction f of HPBW at longer wavelengths. Understandably, the scientists would
like to benefit from the maximum possible imaging quality at these longer wavelengths, and therefore
asked to lower accordingly the ceiling on the error on Im(a), resulting in tighter specs for the aperture
plane alignment at low frequencies. This summarized graphically in Figure 5.
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cc
.
t. a
oin
0.10
"p
0.6
0.10
0.08
0.08
0.06
0.06
0.04
0.04
0.02
0.02
0.00
0
200
400
600
800
0.00
1000 0.00
Amplitude error
0.12
PntAcc/FWHM
0.12
0.05
0.10
0.15
Xs/Rs
Freq
Figure 5. Relation between relative pointing accuracy, amplitude error
(real or imaginary part), and consistent spec for aperture plane alignment.
See text for discussion.
However:

Pointing errors (after pointing corrections) are random, while

The phase across the primary beam is a systematic error that, if stable, can in principle be
calibrated out
Accordingly, and following a preliminary indication that the primary beam phase can indeed be
corrected for, I propose to leave the aperture plane alignment spec at x s / rs  0.09 .
7.
Conclusions
The sensitivity of aperture and focal plane misalignments of ALMA FE optics has been studied. Four
performance criteria have been taken into account: a) loss of on-axis efficiency from aperture plane
misalignment; b) focal plane co-alignment of the two beams of each band; c) aberrations when off-axis
mirrors operate between wavefronts not centered on the foci of the ellipsoid; d) non-uniform phase
across the primary beam resulting from aperture plane misalignment.
It is found that the driving consideration is aperture plane misalignment, in relation with aperture
efficiency. Assuming reasonable values for the various sources of misalignment, it is found that the
performance criteria that were set as goals can be met.
8.
References
[1] Lamb, J. et al. ALMA memo 362. http://www.alma.nrao.edu/memos/
[2] ALMA Project Book. http://www.alma.nrao.edu/projectbk/construction/
[3] Tham, C.Y. and Withington, S.. Receiver Optics Design: Electromagnetic Analysis. Draft report
Sep-2001.
12 / 14
A.1. Perturbation of chief ray from misalignments
Two types of perturbations to perfect alignment are considered: breaks and isolated displacements.
The difference between these two types of misalignments is illustrated in Figure A-1, in the case of an
inline lens system.
Figure A-1. Definition of two types of misalignments.
Note: The displacement (shift+rotation) of an optical element is reckoned in its image (i) space in the
case of a break, and in its object (o) space in the case of a displacement. The center of rotation is
defined in either case at the optical center of the element. These two frames are distinct only in the
case of mirrors, of course.
In the case of a break, the (X,) displacement is propagated through the rest of the system, for
example, in the case of a break after lens/mirror 1:
P
Space( d3). Lens( f2). Space( d2)
where Lens(f) is the usual ABCD matrix for a thin lens/mirror, and Space(d) the matrix for free
propagation over a distance d.
In the case of an individual displacement of an optical element, we need the perturbation of the image
ray as a function of the displacement of that element:
Xi
1
cos(  ) 0 s in(  )
1
i
2
f
0
Xo
. o
Zo
in the case of a mirror (the only case where a Z-displacement along the ray path is significant); and:
Xi
0 0
i
1
f
.
0
Xo
o
in the case of a lens (insensitive to lens tilt in the paraxial approximation); in either case, the
perturbation is then propagated through the rest of the system.
The end result is a 22 matrix P (23 for the individual displacement of a mirror) that relates the X,
displacement of the beam in the focal plane to the misalignment parameters. These matrices have been
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used in section 3.2 to derive the global budget, and in section 4 to study the co-alignment of the two
orthogonal polarizations. The tolerances for individual displacement that would, alone, use up the
alignment tolerance, shown in Figs 2, 3 (section 3.1) were derived from the following equations:
Tol( X )  P2,1
Tol( )  P2,2
Tol( Z )  P2,3
1
1
1
C
C
C
(only in the case of a mirror's individual displacement)
A.2. Loss from phase aberration: amplitude versus power weighting
1. Main reflector, loss of on-axis efficiency.
Let a( x, y ) be the amplitude across the aperture, and let's assume for the unperturbed illumination that
a is real positive (flat phase), and that the perturbed illumination is described by a phase distribution 
with zero mean value. The relative aperture efficiency is given by:
2
Aeff 
 a exp(i) dx dy  a dx dy
A
A
 1   a 2 dx dy
A
 a dx dy
A
2. Arbitrary refocusing mirror, mode coupling
The amplitude coupling coefficient between modes a1 and a2, with phase aberrations is given by:

 a1 a 2 exp( i )



 a1 a1  a 2 a 2
Assume that without the phase perturbation, the mirror gave perfect coupling, i.e. a1  a 2 , and let:
a  a1  a 2 . Then the power coupling amplitude coupling with phase errors is given by:
 a 
  1
2
 a
2 2
2
3. Discussion
A comparison between the two equations for loss of efficiency can be made only when the output
beam is focused at infinity.
In case the phase perturbations have uniform properties over the aperture (magnitude, scale), and the
scale is small w/r to the scale of the field amplitude, the two equations give the same result, which is
1   2 . This is consistent with the fact that, under such conditions, the "main beam" loses amplitude,
but retains its shape; so, the same number measures the loss of on-axis efficiency and loss of mode
(power) coupling.
Differences arise when the scale-length of  is comparable with that of a . Rather than scattering
power into a wide error beam, a more appropriate description is that the main beam is distorted. Then
the decrease in fundamental mode power need not be the same the decrease in on-axis effective area.
When discussing losses at an arbitrary point in telescope optics that is not the main reflector aperture,
on-axis efficiency does not have a clear meaning, and the mode coupling equation is probably more
meaningful.
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